Question

How do you find the slope of $x = 5$?

Answer 1

Here, we will compare the given equation of a line with the standard equation of a line to find the type of line. Then by using the slope of a line using two points formula, we will find the slope of the line. The slope is defined as the ratio of change in the $y$ to the change in the $x$.

Formula Used:
Slope of the line passing through two points is given by the formula $m = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}$ where $\left( {{x_1},{y_1}} \right)$ and $\left( {{x_2},{y_2}} \right)$ are the coordinates of the points respectively.

Complete Step by Step Solution:
We are given an equation of a line as $x = 5$.
We know that the standard equation of a line is $ax + by = c$.
On comparing the given equation to the standard equation of the line, we can say that the value of $x$ is always $5$ because the equation of a line does not have the variable $y$.
So, the equation of a line is independent of the variable $y$.
Thus, clearly, the equation of a line is the equation of a vertical line.
Now, by using the slope of a line through two points formula $m = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}$, we get
$m = \dfrac{{{y_2} - {y_1}}}{{5 - 5}} = \dfrac{{{y_2} - {y_1}}}{0}$
We know that any quantity divided by zero is undefined. So, we have the slope is undefined for the vertical line.

Therefore, the slope is undefined for the equation of line $x = 5$.

Note:
We know that Slope can be represented in the parametric form and in the point form. A point crossing the x-axis is called an x-intercept and A point crossing the y-axis is called the y-intercept. The slope of a line is used to calculate the steepness of a line. We know that the horizontal line does not run vertically i.e., ${y_2} - {y_1} = 0$ , so the slope is zero and also the vertical line does not run horizontally i.e., ${x_2} - {x_1} = 0$, the slope is undefined. These slopes are obtained by using the slope of a line using two points formula.